Paper quantifies neural operators' efficiency for solving nonlinear parabolic PDEs.
problem Quantifying the efficiency of neural operators for solving nonlinear parabolic PDEs.
method Deriving approximation rates by transferring PDEs to integral equations and leveraging Picard's iteration.
result Neural operators can efficiently approximate solution operators of nonlinear PDEs without exponential complexity growth.
Paper analyzes and proves convergence of a new method for solving complex PDEs.
problem Solving high-dimensional nonlinear PDEs and PIDEs with random neural networks.
method Random deep splitting method using random neural networks.
result The method converges to the unique viscosity solution of nonlinear PDEs and PIDEs.
Paper develops techniques to solve complex PDEs involving higher cohomology forms.
problem Develop PDE techniques to study real (p, p) forms on Hermitian manifolds.
method Parabolic approach to establish existence of classical solutions.
result Existence of classical solutions for a large class of fully nonlinear equations.
In this paper we introduce a numerical method for nonlinear parabolic PDEs that combines operator splitting with deep learning. It divides the PDE approximation problem into a sequence of separate learning problems. Since the computational graph for each of the subproblems is comparatively small, the approach can handl…
Study generalizes Picard iteration for nonlinear PDEs, deriving bounds on error.
problem Generalize Picard iteration for nonlinear parabolic PDEs.
method Formulate Picard iteration as abstract state-transition model, derive generalization error bounds.
result Picard depth reduction reduces Picard truncation error without increasing estimation error.
We provide an asymptotic expansion of the value function of a multidimensional utility maximization problem from consumption with small non-linear price impact. In our model cross-impacts between assets are allowed. In the limit for small price impact, we determine the asymptotic expansion of the value function around …
D2SRM solves complex PDEs using deep learning.
problem High-dimensional, Hessian-dependent fully nonlinear parabolic PDEs.
method Single scalar space-time network generating derivative-consistent approximations trained through residuals and penalties.
result Well-posedness and convergence theory established for globally Lipschitz equations.
New theorems prove uniqueness of solutions to geometric PDEs.
problem Proving uniqueness of solutions to geometric PDEs.
method Analyzing nonlinear elliptic PDEs of divergence form.
result Proved several Moser-Bernstein type theorems.
Quantum machine learning solves high-dimensional PDEs with lower variance and improved accuracy.
problem Approximating solutions to high-dimensional parabolic PDEs.
method Pure Variational Quantum Circuit (VQC) for BSDE approximation, using temporal discretization and Monte Carlo simulation.
result VQC achieves lower variance and improved accuracy in most cases, particularly in highly nonlinear regimes.
We propose a new algorithm for solving parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) in high dimension, by making an analogy between the BSDE and reinforcement learning with the gradient of the solution playing the role of the policy function, and the loss functi…
Through the study of some elliptic and parabolic fully nonlinear PDEs, we establish conformal versions of quermassintegral inequality, the Sobolev inequality and the Moser-Trudinger inequality for the geometric quantities associated to the Schouten tensor on locally conformally flat manifolds.
Market illiquidity, feedback effects, presence of transaction costs, risk from unprotected portfolio and other nonlinear effects in PDE based option pricing models can be described by solutions to the generalized Black-Scholes parabolic equation with a diffusion term nonlinearly depending on the option price itself. Di…
New boundary treatment improves accuracy for complex PDEs.
problem Order reduction in high-order IMEX schemes for multidimensional PDEs.
method Novel boundary treatment algorithms for Cartesian meshes, treating implicit-explicit stages similarly to interior points.
result Recovery of designed order of convergence by numerical verification.
There was proposed the method of a factorization of PDE. The method is based on reduction of complicated systems to more easy ones (for example, due to dimension decrease). This concept is proposed in general case for the arbitrary PDE systems, and its concrete investigation is developing for the heat equation case. Th…
Implicit schemes are popular methods for the integration of time dependent PDEs such as hyperbolic and parabolic PDEs. However the necessity to solve corresponding linear systems at each time step constitutes a complexity bottleneck in their application to PDEs with rough coefficients. We present a generalization of ga…
New algorithm solves complex equations using deep learning.
problem High-dimensional nonlinear PDEs and BSDEs.
method Iterated time discretization, deep neural networks, stochastic gradient descent.
result Increased accuracy and reduced complexity compared to existing methods.
In this paper we study a flow by minkowskian curvature where we have a different Minkowski plane at each time. We derive some evolution formulas, present sufficient hypotesis for the short time existence and convexity of solutions and study the motion considering a particular type of families of Minkowski norms. Also, …
Meta-learning base distributions for efficient PDE solutions.
problem Efficiently solving parametric parabolic PDEs across different scenarios.
method Meta-learning base distributions to compute PDE solutions.
result Improves generalization to new parameter regimes.
Paper establishes L∞ estimates for complex Monge-Ampere and Hessian equations.
problem Estimating solutions to complex Monge-Ampere and Hessian equations.
method Uses PDE techniques similar to Phong et al to prove L∞ and Hölder estimates. result Establishes L∞ estimates for both complex Monge-Ampere and Hessian equations. Deep learning model solves high-dimensional PDEs using Actor-Critic approach.
problem Solving high-dimensional nonlinear PDEs efficiently.
method Reformulated PDE into BSDE system, inspired by Actor-Critic algorithm for deep RL.
result Improved model with fewer parameters, faster convergence, and less hyperparameter tuning.
The paper proves well-posedness of nonlocal PDEs related to stochastic control problems.
problem Characterizing equilibrium strategies and value functions for time-inconsistent stochastic control problems.
method Method of continuity and Banach's fixed point arguments, with Schauder prior estimates.
result Global well-posedness of nonlocal fully nonlinear PDEs with sharp a-priori estimates.
New equivalences found linking parabolicity, comparison principle, and capacity on Riemannian manifolds.
problem Understanding parabolicity and related concepts on Riemannian manifolds.
method Establishing new equivalences between parabolicity, comparison principle, and capacity.
result Equivalence between p-parabolicity and the comparison principle for the p-Laplace equation. We present an analytic approach to solve a degenerate parabolic problem associated to the Heston model, which is widely used in mathematical finance to derive the price of an European option on an risky asset with stochastic volatility. We give a variational formulation, involving weighted Sobolev spaces, of the second…
Sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations.
problem Understanding moduli of continuity for fully nonlinear parabolic equations.
method Proving moduli of continuity of viscosity solutions are subsolutions of one-dimensional parabolic equations.
result Sharp Lipschitz bounds and gradient estimates for fully nonlinear parabolic equations with bounded initial data.
The comparison principle for scalar second order parabolic PDEs on functions u(t,x) admits a topological interpretation: pairs of solutions, u1(t,⋅) and u2(t,⋅), evolve so as to not increase the intersection number of their graphs. We generalize to the case of multiple solutions $\{u^α(t,\cdot)\}_{α=1}^…
Tensor trains simplify solving complex PDEs efficiently.
problem Solving high-dimensional parabolic PDEs using traditional methods is computationally infeasible.
method Reformulate PDEs as backward stochastic differential equations and use tensor train format for compression and efficient computation.
result Tensor train methods achieve a good balance between accuracy and computational efficiency.
The paper studies frequency monotonicity for solutions of nonlinear equations under Ricci flow.
problem Frequency monotonicity for positive solutions of nonlinear equations under Ricci flow.
method Obtained parabolic frequency monotonicity for solutions of two nonlinear parabolic equations with bounded Ricci curvature.
result Established integral type Harnack inequalities using parabolic frequency monotonicity.
New framework uses PDE for no-regret generative modeling.
problem Developing efficient generative models for complex distributions.
method Iterative refinement of Brenier maps using mirror gradient descent.
result Converges to optimal Brenier map under various step-size schedules.
The stability and robustness of compact schemes for parabolic PDEs are analyzed.
problem Stability and robustness of compact schemes for solving parabolic PDEs.
method Compact spatial discretization, Crank-Nicolson temporal discretization, eigenvalue analysis of amplification matrix.
result An upper bound on the condition number of the amplification matrix is derived, showing stability.
New method uses neural networks to solve complex PDEs from optimal control theory.
problem Solving high-dimensional Hamilton-Jacobi-Bellman PDEs.
method Iterative diffusion optimization techniques, focusing on path measures and divergences.
result Favourable properties of log-variance divergence for Monte Carlo estimators.
A new method infers parameters from PDEs using Gaussian processes.
problem Estimating unknown parameters in PDEs from noisy data.
method PDE-Informed Gaussian Process (PIGP) method.
result The method bypasses numerical solvers for PDEs and provides uncertainty quantification.
New method uses tensor trains for efficient PDE approximation.
problem High-dimensional PDEs and the curse of dimensionality.
method Tensor trains and backward stochastic differential equations for parabolic PDEs.
result Achieves a favorable trade-off between accuracy and computational efficiency.
We observe that the comparison result of Barles-Biton-Ley for viscosity solutions of a class of nonlinear parabolic equations can be applied to a geometric fully nonlinear parabolic equation which arises from the graphic solutions for the Lagrangian mean curvature flow.
Unified framework solves nonlinear PDEs and IPs using Gaussian processes.
problem Solving and identifying parameters in nonlinear PDEs and inverse problems.
method Gaussian process framework approximating solutions as MAP estimators, reducing to finite-dimensional optimization problem.
result Unified method converges in a small number of iterations for various PDEs.
The paper proves gradient estimates for nonlinear parabolic equations on smooth metric measure spaces.
problem Proving gradient estimates for nonlinear parabolic equations on smooth metric measure spaces.
method Using Souplet-Zhang type estimates and properties of Bakry-Emery Ricci tensor and weighted mean curvature.
result Gradient estimates for nonlinear parabolic equations on smooth metric measure spaces with Dirichlet boundary condition.
Study gradient estimates for nonlinear parabolic equations on Riemannian manifolds.
problem Estimating gradients for nonlinear parabolic equations on Riemannian manifolds.
method Analyzes Fisher-KPP, parabolic Allen-Cahn, and Newell-Whitehead equations on complete noncompact Riemannian manifolds.
result Gradient estimates for positive solutions and Liouville theorem for ancient solutions.
Bayesian methods solve complex nonlinear PDEs efficiently.
problem Solving nonlinear PDEs with high computational cost.
method Bayesian inference with approximate likelihood based on discretization.
result Probabilistic uncertainty quantification for PDE solutions is feasible.
New PDEs of mixed type emerge in fluid mechanics and geometry.
problem Analysis of nonlinear PDEs of mixed type.
method Through historical problems and recent trends.
result Many PDEs are of mixed type, requiring new analysis.
The paper derives new gradient and Hessian estimates for nonlinear parabolic equations.
problem Estimating solutions to nonlinear weighted parabolic equations.
method Derives Li-Yau and Hamilton type gradient estimates, and Hessian estimates.
result New gradient and Hessian estimates for positive solutions of nonlinear parabolic equations.
Unified approach combining BSDEs and PINNs for solving PDEs.
problem Solving high-dimensional partial differential equations.
method Interpolating between BSDEs and PINNs using diffusion loss.
result Unified understanding of numerical approaches for high-dimensional PDEs.
This survey paper is focused on qualitative and numerical analyses of fully nonlinear partial differential equations of parabolic type arising in financial mathematics. The main purpose is to review various non-linear extensions of the classical Black-Scholes theory for pricing financial instruments, as well as models …
Gradient estimates for a parabolic PDE under Ricci-Bourguignon flow on warped product manifolds.
problem Analyzing the Ricci-Bourguignon flow on warped product manifolds.
method Establishing gradient estimates for a parabolic partial differential equation.
result Gradient estimates for the parabolic PDE provide analytic input for geometric applications.
The paper concerns singular solutions of nonlinear elliptic equations, which include removable singularities for viscosity solutions, a strengthening of the Hopf Lemma including parabolic equations, Strong maximum principle and Hopf Lemma for viscosity solutions including also parabolic equations.
This paper includes a proof of well-posedness of an initial-boundary value problem involving a system of degenerate non-local parabolic PDE which naturally arises in the study of derivative pricing in a generalized market model. In a semi-Markov modulated GBM model the locally risk minimizing price function satisfies a…
In his 1954 paper about the initial value problem for 2D hyperbolic nonlinear PDEs, P. Lax declared that he had "a strong reason to believe" that there must exist a well-defined class of "not genuinely nonlinear" nonlinear PDEs. In 1978 G. Boillat coined the term "completely exceptional" to denote it. In the case of $2…
The paper examines solutions to a specific type of nonlinear equation in a disk, proving existence and uniqueness.
problem Existence and uniqueness of radial solutions to a Weingarten equation in a disk.
method Analyzes the linear Weingarten equation in a disk of small radius, considering elliptic, hyperbolic, and parabolic cases.
result Proves existence and uniqueness of radial solutions in the elliptic case, and no solutions in the hyperbolic case.
The paper establishes Harnack inequalities for solutions of nonlinear parabolic equations on manifolds with integral Ricci curvature bounds.
problem Analyzing solutions of nonlinear parabolic equations on manifolds with specific curvature constraints.
method Establishing space-time gradient estimates and integrating them to find Harnack inequalities.
result Harnack inequalities for positive solutions of nonlinear parabolic equations under integral Ricci curvature bounds.
This paper develops a fast algorithm for solving nonlinear PDEs using sparse Cholesky factorization.
problem Efficiently solving nonlinear PDEs with Gaussian processes and kernel methods.
method Sparse Cholesky factorization for near-linear complexity.
result Near-linear complexity algorithm for working with kernel matrices of nonlinear PDEs.